I am stuck with a problem I found in a book on theory of groups. Here it is:
Let $N$ be a normal subgroup of a group $G$ of index $4$. Show (1) that $G$ contains a subgroup of index $2$. Show (2) that if $G/N$ is not cyclic, then there exists three proper normal subgroups $A,B$ and $C$ of $G$ such that $G=A \cup B \cup C$.
This problem arises in a chapter devoted to the homomorphism and isomorphism theorems.